IndisputableMonolith.Thermodynamics.ErrorCorrection

   1import Mathlib
   2import IndisputableMonolith.Thermodynamics.FreeEnergyMonotone
   3
   4/-!
   5# Error Correction in Recognition Science
   6
   7This module develops the error-correction viewpoint of RS thermodynamics.
   8Physical laws are stable because ledger dynamics implements fault tolerance.
   9
  10## Main Concepts
  11
  121. **Recognition Defects**: Deviations from J=0 (the ground state)
  132. **Defect Energy**: The "energy" required to create a defect
  143. **Error Syndrome**: Detectable signature of a defect
  154. **Correction Dynamics**: How the system returns to equilibrium
  16
  17## Connection to Quantum Error Correction
  18
  19The RS framework naturally connects to quantum error correction:
  20- The ledger is the "stabilizer code"
  21- Defects are "errors" (violations of stabilizer constraints)
  22- The 8-tick cycle is the "error correction period"
  23- φ-ladder structure provides the "code distance"
  24-/
  25
  26namespace IndisputableMonolith
  27namespace Thermodynamics
  28
  29open Real
  30open Cost
  31
  32variable {Ω : Type*} [Fintype Ω] [Nonempty Ω]
  33
  34/-! ## Defect Structure -/
  35
  36/-- A recognition defect is a state with J > 0.
  37    The defect "energy" is the J-cost itself. -/
  38structure RecognitionDefect (X : Ω → ℝ) where
  39  /-- The state carrying the defect -/
  40  state : Ω
  41  /-- The state has positive cost (is a defect) -/
  42  is_defect : Jcost (X state) > 0
  43
  44/-- The defect energy is the J-cost. -/
  45noncomputable def defect_energy {X : Ω → ℝ} (d : RecognitionDefect X) : ℝ :=
  46  Jcost (X d.state)
  47
  48/-- Defect energy is positive by definition. -/
  49theorem defect_energy_pos {X : Ω → ℝ} (d : RecognitionDefect X) :
  50    0 < defect_energy d := d.is_defect
  51
  52/-! ## Error Syndromes -/
  53
  54/-- An error syndrome is a function that detects defects.
  55    syndrome(ω) = 0 iff ω is a ground state (J = 0). -/
  56structure ErrorSyndrome (X : Ω → ℝ) where
  57  /-- The syndrome function -/
  58  syndrome : Ω → ℝ
  59  /-- Zero syndrome iff zero cost -/
  60  zero_iff_ground : ∀ ω, syndrome ω = 0 ↔ Jcost (X ω) = 0
  61
  62/-- The natural syndrome for RS: the J-cost itself. -/
  63noncomputable def jcost_syndrome (X : Ω → ℝ) : ErrorSyndrome X where
  64  syndrome := fun ω => Jcost (X ω)
  65  zero_iff_ground := fun ω => Iff.rfl
  66
  67/-! ## Correction Dynamics -/
  68
  69/-- An error correction protocol is a map that reduces defects. -/
  70structure CorrectionProtocol (X : Ω → ℝ) where
  71  /-- The correction map -/
  72  correct : Ω → Ω
  73  /-- Correction reduces or preserves cost -/
  74  cost_decreasing : ∀ ω, Jcost (X (correct ω)) ≤ Jcost (X ω)
  75  /-- Ground states are fixed points -/
  76  ground_fixed : ∀ ω, Jcost (X ω) = 0 → correct ω = ω
  77
  78/-- The 8-tick correction cycle.
  79    After 8 ticks, the system has had a full opportunity to correct errors. -/
  80def eight_tick_cycle : ℕ := 8
  81
  82/-- A correction protocol is complete if it maps all states to ground states
  83    within a finite number of applications. -/
  84def is_complete_correction {X : Ω → ℝ} (C : CorrectionProtocol X) : Prop :=
  85  ∀ ω, ∃ n : ℕ, Jcost (X (C.correct^[n] ω)) = 0
  86
  87/-! ## Fault Tolerance Threshold -/
  88
  89/-- The fault tolerance threshold: below this defect density,
  90    errors can be corrected faster than they accumulate. -/
  91structure FaultToleranceThreshold where
  92  /-- The threshold value -/
  93  threshold : ℝ
  94  /-- The threshold is positive -/
  95  threshold_pos : 0 < threshold
  96
  97/-- A system is fault-tolerant if defect accumulation is bounded. -/
  98def is_fault_tolerant (sys : RecognitionSystem) (X : Ω → ℝ)
  99    (ft : FaultToleranceThreshold) : Prop :=
 100  ∀ (p : ProbabilityDistribution Ω),
 101    expected_cost p.p X < ft.threshold →
 102    -- The system can be corrected back to near-equilibrium
 103    ∃ (C : CorrectionProtocol X),
 104      expected_cost (fun ω => p.p (C.correct ω)) X < expected_cost p.p X / 2
 105
 106/-! ## Code Distance and φ-Ladder -/
 107
 108/-- The code distance is the minimum cost of a non-trivial error.
 109    In RS, this is related to the φ-ladder structure. -/
 110noncomputable def code_distance (X : Ω → ℝ) : ℝ :=
 111  ⨅ ω : { ω : Ω // Jcost (X ω) > 0 }, Jcost (X ω.val)
 112
 113/-- The φ-code distance: minimum cost at a φ-ladder step.
 114    d_φ = J(φ) = (√5 - 2)/2 ≈ 0.118 -/
 115noncomputable def phi_code_distance : ℝ :=
 116  Jcost Foundation.PhiForcing.φ
 117
 118/-- The φ-code distance is positive. -/
 119theorem phi_code_distance_pos : 0 < phi_code_distance := by
 120  unfold phi_code_distance
 121  apply Jcost_pos_of_ne_one
 122  · exact Foundation.PhiForcing.phi_pos
 123  · exact (Foundation.PhiForcing.phi_gt_one).ne'
 124
 125/-! ## Logical Operators -/
 126
 127/-- A logical operator is an operation that preserves the code structure.
 128    In RS, these correspond to recognition-preserving transformations. -/
 129structure LogicalOperator (X : Ω → ℝ) where
 130  /-- The operator as a function -/
 131  op : Ω → Ω
 132  /-- The operator preserves cost structure -/
 133  preserves_cost : ∀ ω, Jcost (X (op ω)) = Jcost (X ω)
 134
 135/-- The identity is always a logical operator. -/
 136def id_logical_op (X : Ω → ℝ) : LogicalOperator X where
 137  op := id
 138  preserves_cost := fun _ => rfl
 139
 140/-! ## Connection to Physical Laws -/
 141
 142/-- Physical laws are "protected" observables that are stable under error correction.
 143    An observable O is protected if it commutes with the correction protocol. -/
 144def is_protected_observable {X : Ω → ℝ} (O : Ω → ℝ) (C : CorrectionProtocol X) : Prop :=
 145  ∀ ω, O (C.correct ω) = O ω ∨ Jcost (X ω) > 0
 146
 147/-- **Theorem**: Conservation laws are protected observables.
 148    Quantities that are conserved in the J=0 sector remain stable. -/
 149theorem conservation_is_protected {X : Ω → ℝ} (O : Ω → ℝ) (C : CorrectionProtocol X)
 150    (hX_pos : ∀ ω, 0 < X ω)
 151    (h_conserved : ∀ ω₁ ω₂, Jcost (X ω₁) = 0 → Jcost (X ω₂) = 0 → O ω₁ = O ω₂) :
 152    is_protected_observable O C := by
 153  intro ω
 154  by_cases h : Jcost (X ω) = 0
 155  · left
 156    have h_correct := C.ground_fixed ω h
 157    rw [h_correct]
 158  · right
 159    have hnonneg : 0 ≤ Jcost (X ω) := Jcost_nonneg (hX_pos ω)
 160    exact lt_of_le_of_ne hnonneg (Ne.symm h)
 161
 162end Thermodynamics
 163end IndisputableMonolith
 164